where \(\varphi \)  is the flow arising from the rough differential equation when \(b\equiv 0\) . This is done in Theorem REF . Somehow this is a way of avoiding the Girsanov transform from Davie's work [1]}, but still working with the solution of the driftless equation. However, this comes at price: we need to work with the flow instead of a trajectory of the driftless system. Furthermore, we can see the appearance of the inverse of the Jacobian \((D\varphi )^{-1}\)  that we need to handle. Another objective of this part is to exploit an idea from [2]} and to exhibit a criterion for this averaged field
\(t,x \mapsto \int _{0}^t \big (D\varphi _t(x)\big )^{-1} b\big (\varphi _t(x)\big )\mathop {}\!\mathrm {d}r\) 
